3 The Regular Conformal Field Equations

It is clear that Penrose’s method of conformally compactifying space-time does provide a convenient and elegant framework for discussing questions of asymptotics. What is not obvious, however, is how to answer the question of existence of space-times that actually do possess the asymptotic structure suggested by the conformal picture. On the one hand, a very specific geometric property of the space-time is required, namely the possibility to attach a smooth conformal boundary, and on the other hand, the Einstein equation, a differential equation on the metric of the space-time, has to be satisfied. It is not clear whether these two different requirements are indeed compatible.

To be more precise, one would like to know how many asymptotically flat solutions of the field equations actually exist, and whether this class of solutions contains the physically interesting ones that correspond to radiative isolated gravitating systems. The only possible avenue to answering questions of existence consists of setting up appropriate initial value problems and proving existence theorems for solutions of the Einstein equations subject to the appropriate boundary conditions.

The conventional Cauchy problem, which treats the Einstein equation as a second-order partial differential equation for the metric field, is already rather complicated by itself (see e.g. the review by Choquet-Bruhat and York [29]). But to obtain statements of the type mentioned above is further complicated by the fact that in order to discuss the asymptotic fall-off properties of solutions one would need to establish global (long time and large distance) existence together with detailed estimates about the fall-off behaviour of the solution.

The geometric characterization of the asymptotic conditions in terms of the conformal structure suggests that we should discuss the existence problem also in terms of the conformal structure. The general idea is as follows: Suppose we are given an asymptotically flat manifold, which we consider as being embedded into an appropriate conformally related unphysical space-time. The Einstein equations for the physical metric imply conditions for the unphysical metric and the conformal factor relating these two metrics. It turns out that one can write down equations that are regular on the entire unphysical manifold, even at those points that are at infinity with respect to the physical metric. Existence of solutions of these “regular conformal field equations” on the conformal manifold then translate back to (semi-)global results for asymptotically flat solutions of the field equations in physical space. This approach towards the existence problem has been the programme followed by Friedrich since the late 1970’s.

In the remainder of this section we will discuss the conformal field equations, which were derived by Friedrich, and the various subproblems that have been successfully treated using the conformal field equations.

 3.1 General properties of the conformal field equations
 3.2 The reduction process for the conformal field equations
 3.3 Initial value problems
 3.4 Hyperboloidal initial data
 3.5 Space-like infinity
 3.6 Going further

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